#StackBounty: #distributions #residuals #rayleigh Proof that $f(t) P(x; sigma = 1)$ is equivilant to $P(x; sigma = f(t))$ where $P(x;…

Bounty: 100

Consider some function or lineshape, $f(t)$, whose form is known. Also consider some one-parameter PDF, $P(x;sigma)$ where $sigma$ is a shape parameter — and importantly defines the mode of $P(x;sigma)$.

Now consider the following case. I now measure some data as a function of $t$ and I know that each measured point is sampled from the PDF $P(x;sigma)$, except now the shape parameter, $sigma$ is defined by $f(t)$. Such that
$$P(x;sigma = f(t))$$
That is, for every value of $t$ I measure my data, I am measuring a data point which is drawn from the PDF $P(x;sigma)$, but with a different shape parameter each time as $sigma = f(t)$.So I have some measured data
$$F(t)sim P(x;sigma = f(t))$$

If I now take the log of both my measured data and function, and subtract them
$$R_{rm{log}} = log_{10}(F) – log_{10}(f(t))$$
i.e. looking at the residuals, and converting back into a linear scale,
$R_{rm{lin}} = 10^{R_{rm{log}}}$, I find that the distribution of my residuals follow the distribution
$$P(x;sigma = 1)$$

How can I prove this mathematically?

I’ll show my scenario with some simulation. It is written in Mathematica but hopefully the code is
fairly readable.

f[A_, w0_, t_] := A * Sin[w0 * t] + 2 * A

I then generate my un-noised pure "fit function". I chose A = 1, w0 = 0.1

UnnoisedData =  Table[Log10[f[1, 0.1, t]], {t, 0 , 10, 10^-3}];

This generates the noised data from t = 0 to t = 10 in a step size of 10^-3. I now generate the noised data as

NoisedData = Table[Log10[RandomVariate[RayleighDistribution[f[1, 0.1, t]]]], {t, 0 , 100, 10^-3}];

Here I have chosen the Rayleigh distribution and in Mathematica, the function RayleighDistribution[s] takes one argument s which is the shape parameter I described above. Notice in both cases I have already taken the logarithm to the base 10.

I know generate my residuals

Rlog = NoisedData - UnnoisedData;
Rlin = 10^Rlog;

If I look at the histogram of my residuals $Rlin$ I see

enter image description here

So my residual in linear units follow a perfect Rayeligh distribution with a shaoe parameter $sigma = 1$. I have confirmed this with large numbers of simulations. This example has $100,001$ data points, and perfomred Maximum Likelihood evaluations to confirm the shape parameter.

Some additional comments

Thinking about the problem a little more, I suppose what it comes down to is how the following expressions are related:
$$f(t) P(x;sigma = 1)$$
$$P(x; sigma = f(t))$$
because if we consider $sigma$ to be the mode of the PDF $P(x; sigma)$, then the two expressions above must be somehow equivalent.

Get this bounty!!!

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